Planets and relativity

I was wondering: Are the stars and their systems in our galaxy stable relative to each other? I mean - how fast do they move in relation to one another. Are there significant time dilation on some planets or systems in our galaxy relative to us?

I'm thinking about this because I was watching Star Trek, and I was wondering if it wouldn't create problems of time determination if different systems have significant time dilation :)

The orbital velocity of stars in a galaxy is pretty much constant regardless of the distance from the galactic centre.http://en.wikipedia.org/wiki/Galaxy_rotation_curve
Which, by the way, is a result that seems to contradict the Keplerian intuition(slower orbital velocity the farther you get - as in e.g.our solar system), and which led to the idea of dark matter.
The orbital speed in Milky Way is roughly 200-250 km/s.

Note that this still means that the galaxy should(intuitively) "wind itself up" into an ever tighter spiral, as the stars closer to the galactic disk's rim have to travel longer distance(2∏R, R=distance from the centre) than the ones closer to the centre, despite having very similar speed.
So, accordingly, stars do not stay in the same places relative to each other. The ones close to the centre overtake those farther away on a regular basis.

Additionally, stars may have some additional velocity unrelated to their orbital speed, due to e.g.close encounters with other stars.

As for the relativistic effects - these velocities are still significantly lower than the speed of light, so the effects are not great (in the order of magnitude of miliseconds per day).
Still, seeing how the relativistic effects cause e.g. clocks on Earth to run differently than those of Earth's satellites( http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html ), it's not completely negligible - especially for precise measurements.

Note that the gravitational relativistic effects(time runs slower near massive bodies) are much greater in the case mentioned above.

Bottom line: disregarding all the other problems that warping around the galaxy might pose, time synchronisation might be necessary upon arrival, but it wouldn't be anything large.